**1. Abstract Set Theory (1953), by A A Fraenkel.**

**1.1. Review by: Robert McNaughton.**

*The Journal of Symbolic Logic* **20** (2) (1955), 164-165.

This book is an intuitive and non-formal, though axiomatic, introductory textbook on set theory, differing considerably from the author's German text. Though devoid of any symbolic logic, its subject matter and quality are such as to make it vital to readers of this Journal. It is "chiefly intended for undergraduates in mathematics, graduate students in philosophy, and high school teachers." In the reviewer's opinion, it will serve well to introduce readers in these categories to fundamental aspects of set theory; many sections will be of service to more advanced readers as well. It is exceptionally well written from whatever point of view it may be judged, but its chief virtue is that it combines intuitive clarity with absolute rigour, always in the right proportion.

**1.2. Review by: Frederick Bagemihl.**

*Mathematical Reviews*.

This is an exposition, "chiefly intended for undergraduates in mathematics, graduate students in philosophy, and high school teachers", of the elementary classical theory of abstract sets (cardinal numbers, order types, and ordinal numbers). Attention is paid "to the foundations of the theory and to matters of principle in general, including points of logical or general philosophical interest." The author adopts a middle course between a naive and an axiomatic development of the theory: "certain principles, similar to Zermelo's axioms, are introduced and at the important turns of the exposition it is pointed out how the constructions required can be based on those principles, while elsewhere the development proceeds without explicit reference to the principles".

**2. Abstract Set Theory (2nd edition, 1961), by A A Fraenkel.**

**2.1. From the Preface:**

In this second edition [1st edition (1953)] the arrangement in general lines has remained unchanged. However, essential changes of detail on almost every page have been made for several reasons. The text underwent a complete revision which renders the first sections more concise; many remarks of secondary importance and references to literature of minor significance were dropped; for some matters of principle the reader is now referred to the author and Y Bar-Hillel's 'Foundations of set theory' (1958). Part of the space saved was used to add new material in the text and the exercises and to utilize pertinent publications which appeared in the decade 1950-1959.

**2.2. Review by: Alonso Church.**

*The Journal of Symbolic Logic* **28** (2) (1963), 168-169.

This book is a very extensively revised edition [of the 1955 edition]. It is an excellent elementary textbook at the undergraduate level, treating cardinal numbers, order types, well-ordered sets, and ordinal numbers.

**3. Abstract Set Theory (3rd edition, 1966), by A A Fraenkel.**

**3.1. From the Preface:**

In this new edition, changes have been limited to modifications in some thirty pages. The principal changes are: first, Paul J Cohen's discoveries regarding the continuum problem (1963) are mentioned in short; second, two remarks of A Church, regarding the equality between sets and the Cartesian product, were taken into account. Furthermore, a supplementary bibliography was added, containing mostly literature which appeared during the last few years, and the Index of Authors was renewed.

**3.2. Review by: P. J. M.**

*The Review of Metaphysics* **20** (2) (1966), 366.

The first edition of this now classical work appeared in 1953, the second heavily revised edition in 1961; this most recent edition is a revision in detail only of the previous one. The book is divided into three parts, the first two dealing with finite and infinite sets, infinite cardinals and their arithmetic, and related remarks on non-standard mathematics and the equivalence of various definitions of finitude. The third part con siders ordered sets and isomorphism types, the special case of linearly ordered (and dense) sets, well-ordered sets in general, the relations of ordinals and cardinals. ... Anyone who works his way through this delightfully written text will come away thoroughly prepared to attack more advanced work in the subject.

**4. Integers and Theory of Numbers (1955), by A A Fraenkel.**

**4.1. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette* **40** (333) (1956), 235.

This is the first of a series of monographs on modern mathematics, based on Fraenkel's talks in the Israel adult education programme; further volumes will deal with modem algebra and transfinite numbers. They are intended for the competent pupil or intelligent layman who may wish to know what modern mathematics is doing. The present volume discusses the integers as cardinals and ordinals, then proceeds to theory of numbers (primes, Fermat's theorems, algebraic numbers and ideals) and ends with a formal abstract account of the extension from the field of integers to the field of rationals. It is gratifying to see how a skilled exposition can cover so much ground with very little technical apparatus, though inevitably some results are merely quoted, while in some instances the proofs are relegated to an appendix. But this ease should not delude the careful reader, who will mark Fraenkel's warning that the treasures of mathematics in this field "may be plucked only by one armed with the weapons of higher analysis and with the abstract and complex methods of modern arithmetic".

**4.2. Review by: William Elliott Jenner.**

*The Mathematics Teacher* **48** (8) (1955), 571.

This book is the first of three intended to give non-professionals some idea of what serious mathematics is about and is directed mainly to high school students, college freshmen, and interested laymen. It deals with cardinal and ordinal numbers, the construction of the ration ale by number pairs, and contains a beautifully written chapter on so-called elementary number theory. Although a popular book, in the literal sense, nevertheless it contains a good deal of real mathematics, and the seriousness and enthusiasm with which it is written could not fail to impress the reader. It most definitely should be in the school library so that young students could get at it.

**4.3. Review by: John Dyer-Bennet.**

*Science, New Series* **122** (3165) (1955), 380.

This volume, the author explains in the preface, is essentially a translation of the first part of his earlier book, Mavo LeMathematika, which was written in Hebrew and grew out of talks given by him, over a period of many years, as part of the adult-education program in Israel. It is to be followed by two more volumes of a similar nature, one on the fundamental concepts of modern algebra, the other on the theory of sets. ... It is an attempt by a mathematician of wide and deep learning to give the intelligent layman some understanding of the nature of our number system and of mathematics in general. It will prove to be a difficult book for such a person, and I shall not try to predict how many there will be who will devote the necessary effort to the task. Those who do, however, will find it an enlightening and stimulating experience.

**4.4. Review by: Harold Davenport.**

*Mathematical Reviews*.

... this little book is mainly concerned with the logical foundations of the natural numbers and the rational numbers. The chapter on the theory of numbers is very elementary and limited in scope, but emphasizes those topics which will be of interest to a wide public. The exposition throughout the book is very clear. Two similar volumes are planned: one on the fundamental concepts of algebra and one on the fundamental concepts of the theory of sets.

**5. Foundations of Set Theory (1958), by A A Fraenkel and Y Bar-Hill.**

**5.1. Review by: Ruben Louis Goodstein.**

*The Mathematical Gazette* **44** (348) (1960), 148-149.

Written for graduate students in mathematics and philosophy this masterly survey of the foundations of mathematics covers not only set theory but the whole range of foundation studies. After a vivid historical introduction describing the paradoxes of set theory there follows an account of the axiomatic foundations of the Zermelo style system which Fraenkel used informally in his introductory volume in the same series, Abstract Set Theory. A central feature of this account is a very full discussion of the axiom of choice. The authors employ a balanced blend of formal and informal methods of presentation, symbolic logic being introduced mainly in the statement of axioms and definitions, to ensure accuracy. ... For its genial style and its appraisal of new work as well as for its accumulation of historical detail this book will be studied with pleasure and profit by a wide circle of readers.

**5.2. Review by: Patrick Suppes.**

*The Philosophical Review* **71** (2) (1962), 268-269.

[The book] affords a leisurely and detailed introduction to the foundations of mathematics, with particular emphasis on set theory. The five chapters are concerned with the antinomies, the axiomatic foundations of set theory, the theory of types, intuitionism, and metamathematical approaches, in this order. An enormous amount of material is discussed, at least cursorily, and a very lengthy bibliography is included.

**5.3. Review by: Joseph R Shoenfield.**

*The Journal of Symbolic Logic* **29** (3) (1964), 141.

This book serves as a complement to Fraenkel's 'Abstract Set Theory' (1955). The latter develops set theory from axioms. The book under review studies the axioms themselves. Hence almost no theorems of set theory are stated or proved. Theorems about axiomatic set theory (e.g., theorems on consistency and relative consistency) are stated, but not proved. On the other hand, almost every axiom system for set theory deserving of any attention is discussed. Philosophical issues arising in axiomatic set theory are discussed quite fully. ... This book has very few prerequisites, and could be a good introduction to set theory for a graduate student, or even a bright undergraduate. For the more experienced reader, the most valuable part will be the discussion of and bibliographical references to the less well-known axiom systems of set theory.

**6. Foundations of Set Theory (2nd edition, 1973), by A A Fraenkel, Y Bar-Hill and A Levy.**

**6.1. Review by: John L Bell.**

*The British Journal for the Philosophy of Science* **26** (2) (1975), 165-170.

Of all the many branches of mathematics, set theory has the closest links with logic and philosophy. Accordingly, an author who attempts to give an adequate discussion of the foundations of set theory is confronted with a formidable task, for he must not only describe the purely technical aspects of the subject, but must also elucidate the role that logic and philosophy have played in its development. In the present book - a revised and expanded edition of the well-known work first published in 1958 - the authors have taken on this task and, as in the earlier edition, produced a clearly written and remarkably comprehensive account of the development and present status of set theory. ... this book is a masterly survey of its field. It is lucid and concise on a technical level, it covers the historical ground admirably, and it gives a sensible account of the various philosophical positions associated with the development of the subject. This work is essential reading for any mathematician or philosopher - professional or prospective - who has an interest in the foundations of mathematics

**7. Mengenlehre und Logik (1959), by Abraham A Fraenkel.**

**7.1. Review by: Presses Universitaires de France.**

*Revue de Métaphysique et de Morale* **65** (3) (1960), 376.

The purpose of this book is not to present in a systematic way, a more or less extended part of set theory, but to present to readers without special mathematical training, problems at the borders of set theory and logic - like those of mathematical infinity or the set theory paradoxes.

**8. Set Theory and logic (1966), by A A Fraenkel.**

**8.1. From the Preface:**

Essentially the booklet is a translation of my book 'Mengenlehre und Logik' (1959). However, I have adapted the text to the needs of the English-reading public, especially of American college students, and have expanded the treatment of a few subjects.

**8.2. Review by: Elliott Mendelson.**

*American Scientis*t **55** (1) (1967), 72A, 74A.

For almost a half-century, the late Abraham A Fraenkel, Professor Emeritus, Hebrew University, Jerusalem, was the leading expositor of the theory of sets. Moreover, in the 1920's, he himself made some important contributions to that theory: i) the discovery of the necessity of extending Zermelo's axiomatic set theory by adding the so called replacement axiom, and ii) a demonstration of the non-provability of the axiom of choice in a set theory with individuals. The present book, a slightly expanded version of 'Mengenlehre und Logik' (published in 1959), is an in formal discussion of some of the high lights of the theory of sets. The word "logic" in the title indicates only that some attention is devoted to definitions and methods of proof. There is no discussion of mathematical logic or of any technical logical problems. ... the book serves its intended audience well.

**8.3. Review by: Thomas Frayne.**

*The Journal of Symbolic Logic* **34** (1) (1969), 112-113.

The book is designed to provide an easy introduction to the problem of infinity and a treatment of logical problems arising at several critical points in set theory for college freshmen in mathematics, logic, and philosophy. It is written for the most part in a discursive style, with lengthy historical and philosophical discussions of the topics treated. Symbolism is avoided, and the more difficult discussions and proofs are set off in small type. ... The book concludes with remarks on the scope and applications of set theory, describing set theory as "a cornerstone in the foundations of many other branches of mathematics and the chief connecting link between mathematics and logic." The book is lucid and stimulating and should for the most part be rewarding reading for its intended audience.